In this paper, we will focus on the low-rank factor analysis models [1]}, [2]}, [3]}, that consider the more general case \({\Psi }\in \mathcal {D}_p^{++}\) , where \(\mathcal {D}_p^{++} = \left\lbrace  {\Sigma } = {\rm diag}({d}), ~{d}\in \mathbb {R}^p_{+*}\right\rbrace \)  denotes the space of positive definite diagonal matrices.
Given this model, the covariance matrix belongs to the space of rank-\(k\)  plus diagonal matrices, denoted as
\(\mathcal {M}_{p,k}=\left\lbrace {\Sigma }={H} + {\Psi },~{H}\in \mathcal {S}_{p,k}^+, {\Psi }\in \mathcal {D}_p^{++}\right\rbrace .\) 
